DYNAMICAL PROPERTIES OF A LESLIE-GOWER PREY-PREDATOR MODEL WITH STRONG ALLEE EFFECT IN

. This paper is devoted to study the dynamical properties of a Leslie-Gower prey-predator system with strong Allee eﬀect in prey. We ﬁrst gives some estimates, and then study the dynamical properties of solutions. In par-ticular, we mainly investigate the unstable and stable manifolds of the positive equilibrium when the system has only one positive equilibrium.


1.
Introduction. In recent years, population models appearing in various fields of mathematical biology have been proposed and studied extensively due to their universal existence and importance [2]. A typical one is the prey-predator model, and such a type model has played the major role in the studies of biological invasion of foreign species, epidemics spreading, extinction/spread of flame balls in combustion or autocatalytic chemical reaction. Many researchers have been interested in the prey-predator models together with several functional responses.
Among the widely used mathematical models in theoretical ecological, the Leslie-Gower prey-predator model plays a special role in view of the interesting dynamics it possesses. The classical Leslie-Gower prey-predator model takes the form [7]   where u and v represent the densities of prey and predator, respectively, parameters µ and β are positive constants. Several ecologists regarded (1) as a prototypical prey-predator system [8,9]. It is known that system (1) has a globally asymptotically stable equilibrium [5].
In most works for prey-predator models, the prey is assumed to grow at a logistic pattern. But in recent years it was recognized that the prey species may have a growth rate of Allee effect, as a result of mate limitation, cooperative defense, cooperative feeding, and environmental conditioning [6,10]. The Allee effect named after W.C. Allee [1], has significant contribution to population dynamics. Allee effect mainly classified into two ways: strong and weak Allee effect [4,12,13]. There is a threshold population level for the strong Allee effect such that the species will become extinct below this threshold population density. However, when the growth rate decreases but remains positive at low population density, it is called the weak Allee effect.
In consideration of the strong Allee effect, in this paper we study the dynamical properties of the following Leslie-Gower prey-predator model with the strong Allee effect in the prey The Allee threshold is b ∈ (0, 1): a strong Allee effect introduces a population threshold, and the population must surpass this threshold to grow ( [3,11,13]). Because of the reaction term and v(t) > 0, the component u(t) may tend to zero and thus the reaction term v(t)/u(t) may be unbounded. Such a bad structure will bring a lot of difficulties in the analysis.
The main purpose of this paper is to study the dynamical properties of the problem (2). Specifically, if (1.2) has no positive equilibrium, the positive solution converges to (0, 0). That is, both the predator and prey will extinct under this situation. Then, if (1.2) has two positive equilibria, we obtain that the two species may persist when the intrinsic growth rate of predator is large. Later in this paper, we focus on the study of the unstable and stable manifolds of the positive equilibrium when the system has only one positive equilibrium. And in such case we find a triangular attraction basin of this equilibrium when the intrinsic growth rate of predator is large. The paper is organized as follows. In section 2, some estimates of global solutions are given. In section 3, we study the dynamical properties of solutions. 2. Existence and estimates of global solution to (2). In this section, we give some estimates for global solutions and verify the population threshold.
If (3) does not hold, then there exists a constant 0 < α < 1 such that v(t) u(t) < α for all t > 0. Therefore, by the second equation of (2), holds.
(ii) The proof is divided into two cases.
Case 2. u 0 = b and v 0 > 0. From the first equation of (2), we get Hence u(t 0 ) < b and v(t 0 ) ≥ 0 for some small t 0 > 0. Using the result of Case 1, we conclude that The proof is completed.
3. Dynamical properties of (2). Obviously, (b, 0) and (1, 0) are nonnegative equilibria of (2). On the other hand, the positive equilibria of (2) has the form ( u, u), where u satisfies The following results concerning with positive equilibria are obvious: (2) has two positive equilibria: For the simplicity of the notations, we denote u = (u, v) and By direct computation, the linearization of G(u) at u = (ũ,ṽ) is
Then w satisfies .
Proof. (i) The proof is divided into two cases. Case 1. u 0 ≤ v 0 and u 0 <ũ 1 . Set where ε > 0 is a small constant. First, we will prove that A is an invariant region. Let Then w satisfies By directly computation, we get This two inequality show that (u(t), v(t)) won't arrive at the boundary of A except (u 0 , v 0 ) ∈ ∂A. Therefore A is an invariant region. Notice (u 0 , v 0 ) ∈ A, we have This leads to lim t→∞ u(t) = 0. Similarly, we can conclude that lim t→∞ v(t) = 0.
(ii) Let G u ( u 2 ) be defined by (7). We have Since A 2 < βũ 2 , the real part of eigenvalues of G u ( u 2 ) are negative if µ > A 2 , which means that (ũ 2 ,ũ 2 ) is locally stable. The real part of eigenvalues of G u ( u 2 ) are positive if µ < A 2 , which implies that (ũ 2 ,ũ 2 ) is unstable.
Proof. (i) We also use the notations in the proof of Theorem 3.1. Let (u, v) be the unique solution of (2). According to βb − 1 − b = −2 √ b, we have k(1, u) ≥ 0. Similar to the proof of Theorem 3.1, we can show that R 1 is an invariant region of (2).
(iii) As u 0 ≤ v 0 , the conclusion (i) asserts v(t) ≥ u(t) for t ≥ 0. This implies The following proof will be divided into two cases.
Case 1. u 0 ≤ v 0 and u 0 <ũ 3 . In this situation, it is easy to see that According to (11), we have that u ≤ u 0 and u ≤ Cu for all t ≥ 0. This leads to lim t→∞ u(t) = 0. Similarly to Theorem 2.1, we can get lim t→∞ v(t) = 0.
In the proof of Theorem 3.3(ii) we see that (u(t), v(t)) converges to (ũ 3 ,ũ 3 ) if (9) holds. In the following we just need to consider the situation that (8) holds. Set , then there exists a triangular region which is a attraction basin of (ũ 3 ,ũ 3 ).
(ii) From Theorem 3.3(iii), we know that (u(t), v(t)) converges to (0, 0) as Consequently, in order to prove the desired conclusion we just need to show that lim In the following, the proof is divided into two cases. .
Combining (21) and (23) we derive Make use of the following Lemma 3.5, we have that there exists a T ≥ 0 such that It follows from (24) and (25) that Making use of the conclusion (i), one can conclude that Γ(λ) is an invariant region of problem (2), and Notice (20) and (23), we get that, for any t ≥ T , Thanks to µ ≤ 1 2 β √ b and (27), it follows from the definition of h(w, t;λ) that Applying (28), (29) and (30), we have Take advantage of (22) and (26), it yields Integrating the first equation of (12) and using (31) we derive This contradiction shows that our assumption (22) does not hold. Thus, lim t→∞ (u(t), v(t)) = (0, 0) by Theorem 3.3(ii).